Before getting to the technical feasibility of moving the ISS, I feel obligated to point out that operating it at L1 or lunar orbit is impractical for a few reasons:
The ISS is designed for the radiation environment of low Earth orbit. Outside of low Earth orbit, without the protection of Earth‘s magnetic field, crew aboard the station will receive much more radiation exposure.
The ISS depends on regular supply missions from Earth. To supply it at L1 or lunar orbit would require much larger launchers, or more frequent launches of very small payloads.
Crew transport to and from the station, likewise, would have to be done on bigger rockets. The Soyuz launcher can't send a crew to L1 or the moon; the Soyuz spacecraft can't get home from there.
The ISS is already cripplingly expensive to operate in LEO; outside of Earth orbit the costs would increase by about an order of magnitude.
Would it be at least possible to boost the station to lunar orbit or at least to an Earth-Moon Lagrange point where minimal station-keeping could used to maintain orbit? How much fuel would be required to get to either of those orbits? What would be involved in transporting that much fuel to the ISS?
The delta-V requirements from LEO to either the Earth-Moon L1 point or the "Lunar Gateway" orbit are pretty similar -- 3.77 km/s for EML1, or 3.63 km/s for Gateway. For EML1, using the rocket equation ...
$$\Delta v = v_\text{e} \ln \frac{m_0}{m_f} = I_\text{sp} g_0 \ln \frac{m_0}{m_f}$$
...with a specific impulse of 302 seconds (corresponding to the Zvezda thrusters of the ISS, but typical of any hypergolic bipropellant thruster), we get an initial to final mass ratio of 3.57 -- that is, you need about 1030 tons of fuel to push the 400 ton ISS to that destination.
To low lunar orbit is 4.04 km/s, which would require a 3.91 mass ratio, you need 1165 tons of fuel per 400 tons of station.
This would obviously require a substantial addition of tankage modules to the ISS -- a huge project in itself, and pushing the dry mass up, requiring still more fuel. Assuming you're launching the tankage and propellant on something like a Proton rocket with around 20 ton payload capacity, I'd say four launches to add the tanks, then 60 more launches to load the propellant. A larger-payload launcher like Falcon Heavy could possibly bring this down to around 25 launches.
With an ion thruster or similar electric propulsion, the fuel requirements come way down. Low-thrust translunar flight requires substantially more delta-v (the Oberth effect is maximized by short, high-thrust burns, which electric thrusters can't achieve) -- around 7km/s to EML1 or 8 km/s to LLO. With a specific impulse of ~4000 seconds (achieved by the NEXT thruster), only around 100 tons of fuel and tankage would be required. This could possibly be done with a medium-lift mission for the tankage and maybe 5 Protons or 2 Falcon Heavies for the propellant.
